First of all, make sure that your login and password are correct.
If the text field shows the "User not found" error, check your other email addresses and aliases, in case you used a different one. You can also look for the GOG account creation confirmation email in your mailboxes, and check what address it was sent to, exactly. If none of the email addresses work, contact our Support team.
If you seem to be using the correct login credentials, but still cannot log in, try these troubleshooting steps:
wipe your browser's cache and cookies, and reload the GOG.com page,
try a different browser altogether (ideally a freshly-installed browser, with no addons/plugins),
add GOG.com to your firewall/antivirus software's whitelist,
contact us if all else fails.
Should you have problems logging in due to the emails with the two-step authentication codes not arriving, please try the following:
Check your inbox and especially your spam folder for an email titled "Two-step authentication" from the email addresses firstname.lastname@example.org or email@example.com.
Try to whitelist the above addresses if you haven't done so already and request a new code.
Make sure you don't close the login window before entering the 2-step code as it can only be used for the login attempt for which it was created.
After that, wait 30 minutes and try again, and if this doesn't help, please contact us directly so we can help you with the issue.